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Approximation Theory

Moduli of Continuity and Global Smoothness Preservation

  • Book
  • © 2000

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Authors:
  1. George A. Anastassiou

    1. Department of Mathematical Sciences, University of Memphis, Memphis, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Sorin G. Gal

    1. Department of Mathematics, University of Oradea, Oradea, Romania

    View author publications

    You can also search for this author in PubMed   Google Scholar

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Table of contents (20 chapters)

  1. Front Matter

    Pages i-xi
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  2. Introduction

    1. Introduction

      • George A. Anastassiou, Sorin G. Gal
      Pages 1-53

  3. Calculus of the Moduli of Smoothness in Classes of Functions

    1. Front Matter

      Pages 55-55
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    2. Uniform Moduli of Smoothness

      • George A. Anastassiou, Sorin G. Gal
      Pages 57-144

    3. LP-Moduli of Smoothness,1 ≤P <+∞

      • George A. Anastassiou, Sorin G. Gal
      Pages 145-169

    4. Moduli of Smoothness of Special Type

      • George A. Anastassiou, Sorin G. Gal
      Pages 171-199

  4. Global Smoothness Preservation by Linear Operators

    1. Front Matter

      Pages 201-201
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    2. Global Smoothness Preservation by Trigonometric Operators

      • George A. Anastassiou, Sorin G. Gal
      Pages 203-210

    3. Global Smoothness Preservation by Algebraic Interpolation Operators

      • George A. Anastassiou, Sorin G. Gal
      Pages 211-230

    4. Global Smoothness Preservation by General Operators

      • George A. Anastassiou, Sorin G. Gal
      Pages 231-249

    5. Global Smoothness Preservation by Multivariate Operators

      • George A. Anastassiou, Sorin G. Gal
      Pages 251-263

    6. Stochastic Global Smoothness Preservation

      • George A. Anastassiou, Sorin G. Gal
      Pages 266-278

    7. Shift Invariant Univariate Integral Operators

      • George A. Anastassiou, Sorin G. Gal
      Pages 279-295

    8. Shift Invariant Multivariate Integral Operators

      • George A. Anastassiou, Sorin G. Gal
      Pages 297-323

    9. Differentiated Shift Invariant Univariate Integral Operators

      • George A. Anastassiou, Sorin G. Gal
      Pages 325-345

    10. Differentiated Shift Invariant Multivariate Integral Operators

      • George A. Anastassiou, Sorin G. Gal
      Pages 347-372

    11. Generalized Shift Invariant Univariate Integral Operators

      • George A. Anastassiou, Sorin G. Gal
      Pages 373-389

    12. Generalized Shift Invariant Multivariate Integral Operators

      • George A. Anastassiou, Sorin G. Gal
      Pages 391-400

    13. General Theory of Global Smoothness Preservation by Univariate Singular Operators

      • George A. Anastassiou, Sorin G. Gal
      Pages 401-427

    14. General Theory of Global Smoothness Preservation by Multivariate Singular Operators

      • George A. Anastassiou, Sorin G. Gal
      Pages 429-450
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Keywords

About this book

We study in Part I of this monograph the computational aspect of almost all moduli of continuity over wide classes of functions exploiting some of their convexity properties. To our knowledge it is the first time the entire calculus of moduli of smoothness has been included in a book. We then present numerous applications of Approximation Theory, giving exact val­ ues of errors in explicit forms. The K-functional method is systematically avoided since it produces nonexplicit constants. All other related books so far have allocated very little space to the computational aspect of moduli of smoothness. In Part II, we study/examine the Global Smoothness Preservation Prop­ erty (GSPP) for almost all known linear approximation operators of ap­ proximation theory including: trigonometric operators and algebraic in­ terpolation operators of Lagrange, Hermite-Fejer and Shepard type, also operators of stochastic type, convolution type, wavelet type integral opera­ tors and singular integral operators, etc. We present also a sufficient general theory for GSPP to hold true. We provide a great variety of applications of GSPP to Approximation Theory and many other fields of mathemat­ ics such as Functional analysis, and outside of mathematics, fields such as computer-aided geometric design (CAGD). Most of the time GSPP meth­ ods are optimal. Various moduli of smoothness are intensively involved in Part II. Therefore, methods from Part I can be used to calculate exactly the error of global smoothness preservation. It is the first time in the literature that a book has studied GSPP.

Reviews

"This monograph is an intensive and comprehensive study of the computational aspects of the moduli of smoothness and the Global Smoothness Preservation Property (GSPP)."

---Zentralblatt MATH

"Going over the introduction, the reader will get an almost full account, without proofs, of the results presented in the monograph… There are many questions for further research arising from this interesting book. Some of them are formulated by the authors, but many more may be born in the mind of the reader."

—Zentralblatt Math.

Authors and Affiliations

  • Department of Mathematical Sciences, University of Memphis, Memphis, USA

    George A. Anastassiou

  • Department of Mathematics, University of Oradea, Oradea, Romania

    Sorin G. Gal

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